Law of cosines

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In trigonometry, the law of cosines is a statement about arbitrary triangles which generalizes the Pythagorean theorem by correcting it with a term proportional to the cosine of the opposing angle. Let a, b, and c be the sides of the triangle and A, B, and C the angles opposite those sides. Then,

This formula is useful for computing the third side of a triangle when two sides' and their enclosed angle's values are known, and in computing the angles of a triangle if all three sides' values are known.

The law of cosines also shows that

iff <math>\cos C = 0 . \;<math> (Since a, b > 0, which is equivalent to C being a right angle. In other words, this is the Pythagorean Theorem and its converse. Although the law of cosines is a broader statement of the Pythagorean Theorem, it isn't a proof of the Pythagorean Theorem, because the law of cosines derivation given below depends on the Pythagorean Theorem.

Table of contents

1 Derivation (for acute angles)

2 Law of cosines using vectors

3 See also

4 External link

Derivation (for acute angles)

Let a, b, and c be the sides of the triangle and A, B, and C the angles opposite those sides. Draw a line from angle B that makes a right angle with the opposite side, b. If the length of that line is x, then <math>\sin C = \frac{x}{a} , \;<math> which equals <math>x=a \cdot \sin C . \;<math> That is, the length of this line is <math>a \cdot \sin C. \;<math> Similarly, the length of the part of b that connects the foot point of the new line and angle C is <math>a \cdot \cos C. \;<math> The remaining length of b is <math>b - a \cdot \cos C. \;<math> This makes two right triangles, one with legs <math>a \cdot \sin C , \;<math> <math>b - a \cdot \cos C , \;<math> and hypotenuse c. Therefore, according to the Pythagorean Theorem:

<math>c^2 = (a \cdot \sin C)^2 + (b - a \cdot \cos C)^2 \;<math>
<math>c^2 = a^2 \cdot \sin^2 C + b^2 - 2 \cdot a \cdot b \cdot \cos C + a^2 \cdot \cos^2 C \;<math>
<math>c^2 = a^2 \cdot (\sin^2 C + \cos^2 C) + b^2 - 2 \cdot a \cdot b \cdot \cos C \;<math>
<math>\sin^2 C + \cos^2 C \;<math> is always 1, so
<math>c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos C \;<math>

Law of cosines using vectors

Using vectors and vector dot products, we can easily prove the law of cosines. If we have a triangle with vertices A, B, and C whose sides are the vectors a, b, and c, we know that: